Non-split geometry on products of vector bundles

It has recently been established that a product bundle, composed of two gauge structures, under some circumstances, possesses a geometry which does not split. Here we provide an educated extension of the above idea to products of many vector bundles with a distinct group structure associated with each factor fibrespace in the splice. The model employs connection I-forms with values in a space product of Lie algebras, and therefore interlaces the various gauge structures in a non-trivial manner. Special attention is given to the structure of the geometric ghost sectors and the super-algebra they possess.